Collective migration has become a paradigm for emergent behaviour in systems of moving and interacting individual units resulting in coherent motion. In biology, these units are cells or organisms. Collective cell migration is important in embryonic development, where it underlies tissue and organ formation, as well as pathological processes, such as cancer invasion and metastasis. In animal groups, collective movements may enhance individuals' decisions and facilitate navigation through complex environments and access to food resources. Mathematical models can extract unifying principles behind the diverse manifestations of collective migration. In biology, with a few exceptions, collective migration typically occurs at a ‘mesoscopic scale’ where the number of units ranges from only a few dozen to a few thousands, in contrast to the large systems treated by statistical mechanics. Recent developments in multi-scale analysis have allowed linkage of mesoscopic to micro- and macroscopic scales, and for different biological systems. The articles in this theme issue on ‘Multi-scale analysis and modelling of collective migration’ compile a range of mathematical modelling ideas and multi-scale methods for the analysis of collective migration. These approaches (i) uncover new unifying organization principles of collective behaviour, (ii) shed light on the transition from single to collective migration, and (iii) allow us to define similarities and differences of collective behaviour in groups of cells and organisms. As a common theme, self-organized collective migration is the result of ecological and evolutionary constraints both at the cell and organismic levels. Thereby, the rules governing physiological collective behaviours also underlie pathological processes, albeit with different upstream inputs and consequences for the group.

This article is part of the theme issue ‘Multi-scale analysis and modelling of collective migration in biological systems’.

集体迁移已成为在移动和相互作用单个单元而导致连贯运动的系统中紧急行为的范例。在生物学中，这些单位是细胞或生物。集体细胞迁移在胚胎发育中很重要，在胚胎发育中它是组织和器官形成以及诸如癌症侵袭和转移的病理过程的基础。在动物群体中，集体运动可以增强个人的决策能力，并有助于在复杂的环境中导航和获取食物资源。数学模型可以从集体移民的各种表现形式中提取出统一的原则。在生物学中，除了少数例外，集体迁移通常发生在“介观规模”，即单位数量从几十个到几千个不等，与统计机制处理的大型系统相反。多尺度分析的最新发展已允许将介观尺度与微观尺度和宏观尺度联系起来，并适用于不同的生物系统。本主题中有关“集体迁移的多尺度分析和建模”的文章汇编了一系列数学模型思想和用于集体迁移分析的多尺度方法。这些方法（i）揭示了集体行为的新的统一组织原则，（ii）揭示了从单一迁移到集体迁移的转变，并且（iii）使我们能够定义细胞和有机体组中集体行为的异同。作为一个共同主题，自组织的集体迁移是细胞和有机体水平上生态和进化制约的结果。

本文是主题问题“生物系统中集体迁移的多尺度分析和建模”的一部分。